quantum field theory - How do I define time-ordering for Wightman functions?

This is a follow-up question to What are Wightman fields/functions

Ok, so based on my reading, the field operators of a theory are understood to be operator-valued distributions, that is, to be integrated over a smooth function: e.g. $\hat\phi(f)≡\int dx\, f(x) \hat\phi(x)$. But if the smooth function has support over such a large region in space-time that if I try to compute time-ordered products, I run into some ambiguity. How do I define the time-ordering symbol for Wightman functions?

Answer

@ArnoldNeumaier and @dushya have both pointed out correct solutions, but I want to elaborate a bit.

The easy approach is the one dushya suggested. (You can also do what Arnold Neumaier suggests: First define the time-ordered product of operator-valued distributions, and then take expectation values.)

Begin by recalling how Wightman functions are defined. The correlation function of $n$ smeared fields is a multilinear functional $(f_1,...,f_n) \mapsto \langle vac | \hat{\phi}(f_1) ... \hat{\phi}(f_n)|vac\rangle$. You can use the Schwarz nuclear theorem to prove this VEV has a kernel $W$ function such that

$\langle vac |\hat{\phi}(f_1) ... \hat{\phi}(f_n)|vac\rangle = \int f(x_1)...f(x_n) W(x_1,...,x_n) dx_1...dx_n$

This $W$ is a Wightman function. Morally, $W$ is $\langle vac | \hat{\phi}(x_1)...\hat{\phi}(x_n)|vac\rangle$.

Now, if you want to define a time-ordered correlation function, all you have to do is permute the arguments in the Wightman function.

There's a somewhat more conceptual approach available, via analytic continuation from Minkowski signature to Euclidean and then back.

You can prove that the Wightman functions $W(x_1,...x_n)$ are boundary values of an analytic function defined on (a domain in) the product of $n$-copies of the complexification of Minkowski space. If you restrict this analytic function to the Euclidean subspace, the function you get -- known as a Schwinger function -- is permutation-invariant. (It has to be, because it's the output of a Euclidean functional integral constructed with only commuting variables.)

So where did the ordering of operators go? It's in the analytic continuation. The Schwinger functions are analytic, but not entire. The analytic continuation of a Schwinger function has one branch for each of the possible orderings of the arguments. So you can get the time-ordered correlation functions by continuing the Wightman functions from Minkowski space to Euclidean space and then looping back along a different branch.

The basic picture above -- Wightman functions and their analyicity -- is explained beautifully in Streater & Wightman's book PCT, Spin, Statistics, and All That. It's also in the 2nd of Kazhdan's IAS lectures. (Although the heavy lifting needed to show that the time-ordered Wightman functions are well-defined seems to be in Epstein & Eckmann's Time-ordered products and Schwinger functions.) It's also used regularly in particle physics texts, when computing propagators in momentum space. (Have a look at the contour integrals in Peskin & Schroeder's discussion of the Klein-Gordon propagator.)